To introduce the concept of pre-LMO, it is necessary to briefly recapitulate the F2M scheme[1]}, [2]}, [3]}, [4]} for constructing ROHF LMOs.
It involves two steps: (1) generate a set of orthonormal primitive fragment localized molecule orbitals (pFLMO) \(\lbrace \phi _{\mathrm {c}}^{\mathrm {pFLMO}},\phi _{\mathrm {a}}^{\mathrm {pFLMO}},\phi _{\mathrm {v}}^{\mathrm {pFLMO}}\rbrace \) 
from subsystem SCF calculations and localizations (for more details, see Ref.[1]}, [2]});
(2) perform a least-change type of block-diagonalization of the converged Fock matrix represented in the pFLMO basis. Since the pFLMOs
are already classified into doubly occupied (denoted by c), singly occupied (denoted by a) and unoccupied (denoted by v), the ROHF equation takes the following form
\(\begin{pmatrix}\mathbf {F}_{\mathrm {cc}} & \mathbf {F}_{\mathrm {ca}}&\mathbf {F}_{\mathrm {cv}} \\\mathbf {F}_{\mathrm {ac}} & \mathbf {F}_{\mathrm {aa}}&\mathbf {F}_{\mathrm {av}} \\\mathbf {F}_{\mathrm {vc}} & \mathbf {F}_{\mathrm {va}}&\mathbf {F}_{\mathrm {vv}}\end{pmatrix}\begin{pmatrix}\mathbf {C}_{\mathrm {cc}} & \mathbf {C}_{\mathrm {ca}}&\mathbf {C}_{\mathrm {cv}} \\\mathbf {C}_{\mathrm {ac}} & \mathbf {C}_{\mathrm {aa}}&\mathbf {C}_{\mathrm {av}}\\\mathbf {C}_{\mathrm {vc}} & \mathbf {C}_{\mathrm {va}}&\mathbf {C}_{\mathrm {vv}}\end{pmatrix}=\begin{pmatrix}\mathbf {C}_{\mathrm {cc}} & \mathbf {C}_{\mathrm {ca}}&\mathbf {C}_{\mathrm {cv}} \\\mathbf {C}_{\mathrm {ac}} & \mathbf {C}_{\mathrm {aa}}&\mathbf {C}_{\mathrm {av}}\\\mathbf {C}_{\mathrm {vc}} & \mathbf {C}_{\mathrm {va}}&\mathbf {C}_{\mathrm {vv}}\end{pmatrix}\begin{pmatrix}\mathbf {E}_{\mathrm {c}} & \mathbf {0} &\mathbf {0} \\\mathbf {0} & \mathbf {E}_{\mathrm {a}}&\mathbf {0}\\\mathbf {0} & \mathbf {0} &\mathbf {E}_{\mathrm {v}}\end{pmatrix}. \) 
